Projections and orthogonality
Problem 9.263 · hard
Write \( \displaystyle \mathbf u = \left\langle -4, 5, -6 \right\rangle \) as the sum of a vector parallel to \( \displaystyle \mathbf v = \left\langle -6, -1, 3 \right\rangle \) and a vector orthogonal to it.
- \[ -18 - 5 + 24 = 1 \]u·v.✓ Proved
- \[ 1 + 9 + 36 = 46 \]‖v‖².✓ Proved
- \[ \left[\begin{matrix}- \frac{3}{23}\\- \frac{1}{46}\\\frac{3}{46}\end{matrix}\right] \]proj = (u·v/‖v‖²) v.✓ Proved
- \[ \left[\begin{matrix}- \frac{89}{23}\\\frac{231}{46}\\- \frac{279}{46}\end{matrix}\right] \]The orthogonal part is u − proj.✓ Proved
- \[ 0 \]It is orthogonal to v.✓ Proved
Answer \( \left\langle - \frac{3}{23}, - \frac{1}{46}, \frac{3}{46} \right\rangle + \left\langle - \frac{89}{23}, \frac{231}{46}, - \frac{279}{46} \right\rangle \)
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the remainder is orthogonal to v and the projection is parallel to it |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the vector projection formula and verifies orthogonality. The arithmetic is correct, and the final decomposition matches the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the vector projection formula and verifies orthogonality. The arithmetic is correct, and the final decomposition matches the stated answer.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the vector projection formula to decompose u into parallel and orthogonal components. All arithmetic checks out, and the orthogonality condition is verified.gpt-oss:20b: pass 2026-10-05
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/vector_projection, checked 2026-10-05 with SymPy 1.14.0.